Contents

Abstract

One aspect of the Io-Jupiter interaction studied by many authors
is the generation of Alfvén waves by Io
moving in the magnetized
plasma. In our study, we focus on an additional mechanism of the
interaction between Io and Jupiter based on MHD slow shocks
propagating from Io toward Jupiter along a magnetic flux tube.
These MHD slow shocks are produced by plasma flow injected by Io,
which is considered as a source of ionized particles. The
propagation of the slow shocks is calculated along a given
magnetic flux tube from Io to Jupiter. The total pressure is
assumed to be a known function of the distance measured along the
tube. It is determined as the magnetic pressure corresponding to
the undisturbed Jovian magnetic field calculated in a dipole
approximation. The material coordinates are used to describe the
plasma flow along the magnetic tube. The peculiarity of this
problem stems from the fact that the total pressure increases by a
factor of
105, whereas the cross section of the magnetic flux
tube decreases by a factor of 300. The result is that the plasma
velocity after the shock front substantially increases toward
Jupiter with increasing magnetic pressure. The electric potential
difference along the magnetic field is estimated, which is
produced by the accelerated plasma flow propagating with the MHD
slow shocks.

Introduction

In 1954,
Burke and Franklin [1955]
discovered radio emissions from
Jupiter at 22.2 MHz. Subsequent observations established the
strong control of these decametric emissions (DAM) by the
satellite Io. Considerable theoretical and experimental attention
has been given to explain this strong control. These
examinations are based on the global interaction between Io and
Jupiter. It is generally believed that the main factors of the
plasma torus-Io-Jupiter electrodynamic interaction are Alfvén
wings which are standing Alfvén waves
attached to the satellite
[Bagenal and Leblanc, 1988;
Neubauer, 1980].
Slow waves are
expected to be much less intensive and therefore these
magnetohydrodynamic (MHD) waves
received less attention although they have been investigated in
several publications
[Kopp, 1996;
Krisko and Hill, 1991;
Linker et al., 1991;
Wright and Schwartz, 1990].
Unfortunately, these
studies were restricted to the vicinity of Io, whereas the most
important and interesting effects are expected when a slow wave
propagates into the region with strong magnetic field above the
ionosphere of Jupiter. The purpose of this paper is outlined as
follows: We follow a slow wave traveling from its source point at
Io toward Jupiter, estimate the consequences of this propagation
process along the Io flux tube, and finally find some possible
application for the DAM radiation.

There are at least two possibilities how the slow mode waves can
be excited in the vicinity of Io. The first one is connected with
the high volcanic activity of Io. Direct volcanic plumes at Io
cannot supply the bulk of the torus plasma, because the
characteristic plume speed is much less than the escape velocity
(2.6 km s
-1 )
[Hill et al., 1983].
Nevertheless, small portions of
volcanic material are stripped by a cometary type of interaction
with the plasma in the Io torus. Observations show that the plasma
density in the Io torus remains roughly constant, although the
volcanic activity on Io is sporadic. Probably there is a feedback
mechanism due to which an increase of the mass in the plasma torus
causes a corresponding increase in the loss
[Brown and Bouchez,1997].
However, the details of this process are still poorly
understood, the characteristic relaxation times are unknown, and
so far, it is difficult to calculate the increase in the plasma
pressure produced by a concrete volcanic eruption. The generation
of kinetic Alfvén waves produced by volcanic
eruption at Io has
been recently investigated by
Das and Ip [[2000].

Figure 1

The second possibility for the development of a pressure pulse
might be the flow of the torus plasma around Io. Analytical and
numerical studies show a rather complicated picture of the gas
pressure distribution with firstly an enhancement followed by
rarefaction, but slow mode wings are clearly seen
[Kopp, 1996;
Linker et al.,1991].
The gas pressure enhancement in the maximum is
relatively small (~30%) and increases with mass loading
[Kopp, 1996].
As noted, the exact mass loading rate is rather
uncertain. Therefore it is a good idea to use experimental data
for the estimation of the pressure pulse intensity. These
experimental data are provided by the Galileo spacecraft. Figure 1
shows a direct observation of the plasma pressure in the vicinity
of Io at a closest approach of about 0.5
R Io (900 km)
[Frank et al.,1996].
As can be seen, the pressure is enhanced by a
factor ~2-3. However, after this increase, the spacecraft
crossed the cold ionosphere of Io, and thus the data points are
not valid for the warm plasma in the torus. Extrapolating the
increasing curve with a Gaussian function reveals that the real
enhancement of the gas pressure must be in the range ~6-8,
as Figure 1
shows.

Moving along its orbit, Io is followed by a wake of disturbed
plasma pressure. In the frame of Io, these wings look like a
steady-state structure.
However, in a frame of a given magnetic flux
tube passed by Io, the plasma perturbations are not steady: The
plasma pressure is a function of time.

The background plasma parameters of the magnetic flux tube are
considered to be in equilibrium. This equilibrium is reached after
some relaxation time, when a new portion of plasma injected into
the tube is precipitating due to the loss process. This relaxation
time should be much less than the period of the Io motion along
its orbit. The latter condition seems to be valid for the
Io-Jupiter interaction.

In the following, we suggest that there is a positive pressure
pulse of an amplitude of about 6 in a flux tube and calculate the
slow mode propagation toward Jupiter. From the physical point of
view, crossing of the fresh Io flux tube is similar to an
explosion in this Io flux tube, but an explosion of a very
specific type. The peculiar feature of this explosion stems from
two basic facts. First, the slow mode wave is guided along the
magnetic field
(one-dimensional (1-D) explosion), and second, the slow wave propagates inside
a dipole flux tube with progressively decreasing cross section.
For the Io flux tube ( L 6 ) the cross section of the
tube decreases 380 times within a distance of
7.13 R J,
and in addition, the magnetic pressure increases
1.5 105 times. As a result, the flow
velocity has to increase toward
Jupiter rather than to decrease, as it usually happens after a
regular explosion.

So, a scenario that is justified in this paper can be described
as follows (see Figure 2): A pressure pulse produced near Io
generates two slow waves propagating along the Io flux tube into
the southern and northern ionosphere of Jupiter. These slow waves
are quickly converted into nonlinear waves due to a steepening
mechanism with a supersonic flow behind the shock front. The flow
velocity behind the shock increases in the course of the
propagation to Jupiter and reaches values of the order of the
initial Alfvénic velocity (~150 km s
-1 )
at the site
of Io. In its turn, the plasma flow streaming along the Io flux
tube has to generate a field-aligned potential difference due to
the Alfvén mechanism
[see
Serizava and Sato, 1984],
which can be
as large as 1 kV. Therefore the slow mode wave mechanism
seems to contribute to the Io-controlled aurora and radio
emissions together with the generally accepted Alfvén
wings
model.

Mathematical Formulation of the Problem

Figure 2

From the mathematical point of view we have to simulate a
local
explosion inside the Io flux tube. The geometrical situation of
the problem is illustrated in Figure 2.

To describe the perturbations of the magnetic field and the plasma
parameters, we basically apply the system of ideal MHD equations
without dissipation

(1)

(2)

(3)

(4)

(5)

Here
r,
V,
P, and
B are the mass density, bulk
velocity, plasma pressure, and magnetic field, respectively. The
quantity
P denotes the total pressure (the sum of magnetic
and
plasma pressures),
P = P + B2/(8p),
and
k is the
polytropic exponent.

Considering the case of low plasma beta, we assume the total
pressure to be approximately equal to the dipole magnetic
pressure. Therefore in this approximation the total pressure is a
known function of the distance
S along the tube.

From the momentum equation (1),
we obtain the equation for
the field-aligned velocity component

(6)

where
S is the distance measured along the tube.

The conservation law for the magnetic flux results in a new
equation for the magnetic field, i.e.,
BF = Y, where
Y is the magnetic flux, which is constant along
the flux
tube, and
F is the cross section of the magnetic flux tube. For
an ideally conducting plasma, the magnetic field is frozen-in and
the magnetic flux is not a function of time for a given flux tube.

The mass conservation law applied to the thin tube yields the
continuity equation in the following form

(7)

Substituting the cross section expressed through the magnetic
field strength, we obtain the equation

(8)

The system of relations (6)-(8)
is closed by the
entropy equation

(9)

We normalize the magnetic field and the plasma parameters as
follows:

(10)

Here
R J is the Jupiter radius,
VA0 is the Alfvén
velocity, and
r0,
B0 are the mass density and magnetic
field strength in vicinity of Io. The undisturbed plasma
parameters along the magnetic field line are considered to satisfy
the equilibrium equation

(11)

where
G is the gravitational constant,
m is the average mass
of particles,
M is the mass of Jupiter,
r is the radial
distance from the center of Jupiter,
y is the distance to the
rotational axis,
k is the Boltzman constant,
T is the
temperature assumed to be constant along the tube, and
W is
the angular speed of Jupiter, respectively.

The plasma parameters at the Io orbit used in our calculations are
the following
[Neubauer, 1998]:

(12)

These parameters correspond to an Alfvén
speed of
Va = 150 km s
-1.
Similar parameters were used also by
Combi et al. [1998]
for the numerical simulation of mass loading in the vicinity of
Io.

Next, we introduce the material coordinate
a, which is
determined as follows:

(13)

In accordance with the equation above, the quantity
a is
constant along the trajectory of the particle.

From the definition of
a, a differential relation is
obtained

(14)

This is the partial derivative with respect to time under a
constant
a.

Using equations (8), (13), and (14),
we obtain a relation between
a and
B

(15)

This coordinate
a is a particular case of the so-called
frozen-in coordinates introduced by
Pudovkin and Semenov [1977]
for ideal magnetohydrodynamics.

Using the independent coordinate
a instead of
S, we
finally obtain the system of equations

(16)

(17)

(18)

(19)

(20)

For the computation of the shock discontinuities, the adiabatic
equation is not appropriate, because the entropy has a jump at the
shock. Therefore to calculate the shock fronts, the
adiabatic equation has to be replaced by the more general energy
equation:

(21)

To solve the problem, we use a two-step differential numerical
method with a right-angled grid. At the first step, the method of
characteristics is used to calculate the density, velocity, and
magnetic field in the intermediate grid points labeled with
half-integer numbers. Along the characteristics we have the
following equations:

where
Q = B/r, and

For the second step, the plasma parameters and the magnetic field
are calculated in the main grid points labeled with integer
numbers

Here the different signs correspond to the intermediate points
i 0.5 and
t + 0.5 , for the parameters marked by
and
.

The gas pressure and density are calculated from the system of
algebraic equations

This variant of the differential scheme is appropriate for the
computation of plasma parameters in the tube where the total
pressure is not very large compared to the plasma pressure. It is
important to note that the total pressure varies along the
magnetic flux tube from Io to Jupiter by a factor of
1.5 105. In the case of very large
magnetic pressure, the plasma
pressure obtained from the equation for the total pressure, is the
difference between two very large quantities, the total and the
magnetic pressures. This brings about an inaccuracy in the
numerical solution.

Therefore the entropy equation is more convenient for the
computation of the plasma pressure in a region of very large total
pressure. In such a case, the shock front must be separated,
thereby using the jump conditions for the slow shock.

In our calculations, the energy equation together with equations
(16)-(20) are only used for the description of the first
stage of the formation of the slow shock, produced by the local
enhancement of the plasma pressure near Io. After that, when the
shock front is formed, we calculate the propagation of the slow
shock along the magnetic flux tube toward Jupiter by separating
the shock front and using the system with the entropy equation
behind the shock.

Results of MHD Simulation

Figure 3

The first stage of the slow mode wave generation is shown
in
Figure 3.
It can be clearly seen that the initial pressure pulse
is quickly divided into two waves propagating along the flux tube
in opposite directions. The amplitudes of these waves are
decreasing in the course of time, the leading fronts getting more
and more steep, and eventually slow waves are converted into
shocks.

The initial stage is very similar to a classical 1-D explosion, and
the behavior of all parameters (pressure, density, velocity) is
also quite analogous. Thus if the tube cross section would not
change any more, the process of the wave propagation could be
easily predicted: Slow shocks would travel along the flux tube
and be gradually damped with decreasing flow velocity behind the
shock fronts.

Figure 4

Figure 5

However, the flux tube cross section is inversely proportional to
the magnetic field strength and therefore has to decrease as
r3 due to the dipole field configuration. Hence the plasma
flow has to move into a more and more narrow flux tube, and
in
addition, the total pressure is increasing even more rapidly as
r-6. As a result, the wave amplitude firstly stays roughly
constant as long as damping, due to the expansion, is balanced by the
narrow channel effect. After that, the tube cross section starts
to decrease so rapidly that the wave amplitude begins to enhance,
as Figure 4
shows.
All these three stages are especially evident in Figure 5 where
the positions of the maximum values of the plasma pressure,
density, and velocity are shown. The first stage, corresponding to
a classical explosion, is rather small, restricted to the
proximity of Io. It can be seen in numerical simulations of the
plasma torus flow around Io
[Kopp, 1996;
Linker et al., 1991].
However, in their simulations the pressure perturbations for the
compression (there are also rarefactions that we do not consider
here) are weaker than our pressure quantity. This discrepancy
stems from the fact that the amplitude of the pressure variation
is strongly dependent on the mass loading rate, which is poorly
known so far. Strong plasma pressure perturbations are related
with a large mass loading rate
[Combi, 1998;
Kopp, 1996].
In our
simulations, we use the initial pressure variation amplitude of
the factor 6 from the Galileo observations
[Frank et al., 1996].

The second stage is characterized by approximately constant
parameters and is the most prolonged one. Nearly all the way from
Io to Jupiter expanding, damping, and narrow channel effects
effectively compensate each other.

By the arrival at the point ( S 6.3 R J ), the
plasma velocity
reaches its maximum value
0.9 VA0. After the maximum
point, the velocity starts to decrease near Jupiter because of the
enhancement of the background pressure due to the gravitational
force.

As it was shown, an explosion in a flux tube with an increasing
magnetic field and correspondingly decreasing of the tube cross
section is considerably different from an usual 1-D explosion. The
narrow channel effect leads to an intensification of the
propagating wave rather than to damping due to its expansion. This
effect might be important not only in the case of Io but also for
all other situations on the Sun or on the other planets where
pressure pulses can be produced inside a thin flux tube.

Field-Aligned Electric Field

As we saw, pressure pulses created near Io eventually generate
slow shocks accompanied by an accelerated plasma flow behind the
shock. It is known that a supersonic flow produces a
field-aligned electric field
[Serizava and Sato, 1984].
It is a
fact that if the mass velocity of ions is much bigger then the
thermal velocity (supersonic flow), then most of the ions must
have small pitch angles, whereas the electrons, for which the
thermal velocity is much greater, should not be noticeably
disturbed. As a result, the mirror points of the streaming protons
and electrons are located at different positions along the flux
tube leading to a charge separation and the occurrence of a
field-aligned electric field. To calculate the potential
difference, we use the method of
Serizava and Sato [1984],
modified
for the Io flux tube with its very special types of ions. This
method has been checked with numerical simulations
[Schriver, 1999],
and both results coincide within
5%, which is accurate
enough for our purpose.

As a starting point for the mathematical analysis, we assume the
different plasma components at the initial position to be modeled
by the so-called parallel beam distribution function given as

(22)

where
N,
m,
v,
T, and
V refer to particle number
density, mass, velocity, temperature in energy units, and flow velocity,
respectively. Subscript
j denotes the particle species and
subscripts
p and
n characterize the parallel and normal
components of the relevant quantities with respect to the magnetic
field.

In the following, we explicitly use the particle conservation
along the magnetic flux tube and assume that there is no
field-aligned current present. In other words, the flux carried
by the reflected electron contribution
Fe has to be equal to
the reflected ion flux
Fi,s. Thus we have

(23)

where
s denotes the different kinds of ions. This current-free
condition together with the initial charge neutrality lead us to

(24)

where subscript zero refers to quantities at the initial position.

As a next step, we evaluate the respective reflected fluxes
carried by the different species of particles. Therefore we start
with the consideration of the magnetic moment
m,
characterizing the particle's perpendicular velocity. This
quantity is conserved as the particles precipitate toward higher
magnetic field strengths. Hence assuming a dipole magnetic field
configuration leads us to

(25)

where
l denotes the magnetic latitude, and we introduced
the parameter
g referring to the ratio of the magnetic
fields.

Additionally, from the conservation of energy we have

(26)

where
F denotes the electric potential difference, which
is
initially zero, and
q refers to the charge. Substituting
relation (25) into (26) yields

(27)

It is reasonable that at the mirror points the parallel component
of the particle velocity vanishes. Thus the right-hand side of
expression (26) determines a curve in the velocity space
separating the reflected particles from those passing through. The
corresponding fluxes are now derived by calculating the first
moment of the distribution function (22), with regard to
the particular contributions, which carry the reflected flux. The
result is as follows

(28)

(29)

with

(30)

(31)

(32)

where
Erf[ x ] is the error function. The
current-free condition (23) permits to solve
expression (28) together with (29) numerically
to obtain the potential as a function of the distance
from the equatorial plane along the flux tube
S normalized to
the Jupiter radius
R J.

As the electrons pass the developed potential they pick up the
potential drop and are effectively accelerated. The electron
velocity can be determined by

(33)

Figure 6

To simulate the conditions in the Io flux tube, we assume
the plasma stream to contain hot anisotropic ions (S
+ and
O
+ ) with
Tn,i = 5 and
Tp,i = 200 eV,
and isotropic electrons with
Tn,e = Tp,e = 150 eV,
following thereby
[Mei et al., 1995].
The electric potential calculated for the different plasma
velocities and the ratio of the current and thermal speeds of
electrons accelerated by the potential difference are shown in
Figure 6
as functions of the distance along the magnetic flux
tube.

We can summarize the results of our investigations as follows:
First, we note that the potential tends to saturate toward a
maximum value. The prevailing quantities determining the strength
of this maximum potential are the flow speed and the composition
of the ion population. As shown in Figure 6, the potential
increases with the flow speed. In fact, the strength of the
saturation potential is proportional to the flow energy of the ion
contribution and therefore increases with the square of the ion
flow velocity. Now, the influence of the mass of the streaming ion
populations can be clarified. Heavy ion constituents lead to an
enhancement of the potential drop, whereas light ion populations
do not effectively contribute to the potential difference. For our
purpose, the ion population is assumed to consist of heavy sulfur
and oxygen, which leads to a considerable high potential drop.

The main result of this study is that the amount of the maximum
potential difference can be of the order of 1 kV. As the
precipitating electrons pass through the developed potential they
pick up the potential drop and are effectively accelerated to
energies of the order of the strength of the potential drop. The
electron velocity can be determined by
ve 2 qeF / me. These energetic
electrons play a
crucial role in the explanation of aurora on Jupiter and the DAM
radio emissions
[Hill et al., 1983;
Wu, 1987].

Discussion

In this paper we tried to emphasize the role of slow mode waves,
which are not only important in the course of the torus plasma
flow around Io
[Kopp, 1996;
Krisko and Hill, 1991;
Linker et al., 1991;
Wright and Schwartz, 1990]
but can also be
responsible for some specific phenomena such as aurora or DAM
radiation together with Alfvén waves.
The latter mechanism is
much more powerful, and as far as aurora is concerned, we suggest
the following interpretation: Direct observations of the Io
footprint aurora show that there is a bright leading point
corresponding to the projection of Io. In addition, a diffuse
fainter emission is observed, which is extended in longitude with
several bright spots in the tail
[Connerney et al., 1999].
The leader is certainly connected with the first Alfvén
wave
arrival at the Jovian ionosphere. The trailing spots have been
interpreted as arrivals of reflected Alvfén
waves
[Connerney et al., 1999].
Our point is that one of these bright spots
in the tail might be connected with the arrival of the slow shock.
The DAM emissions are also believed to be caused by Alfvén
wings
having their source near the instantaneous Io flux tube
[Bagenal,1983;
Menietti and Curran, 1990].
However, studies of
Queinnec and Zarka [1998]
show that some parts of the DAM emission, in
particular the Io-B radiation, have a
30o-
50o lag
of the source field line and the instantaneous Io flux tube for
the maximum emission frequency. This lag would require an
unrealistically increasing plasma density (more than 10 times
higher than that used by
Bagenal [1983])
to explain the
propagation time of Alfvén waves.

Our study shows that the role of this carrier can be played by
nonlinear slow wave. As it was pointed out, the consequence of the
slow shock propagation is a strong plasma flow behind the shock
front, which in turn leads to a field-aligned electric potential
difference of the order of 1 kV. The nonlinear wave is much slower
than an Alfvén wave and gives the needed
longitude lag in the
range of 30o-50o. Therefore we believe that the
slow wave mechanism can also be responsible for some parts of the
DAM emissions.

Acknowledgments

This work is supported
by the INTAS-ESA project
99-01277, by the Austrian "Fonds zur Förderung
der
wissenschaftlichen Forschung" under project
P12761-TPH, by grant 98-05-65290 from the Russian
Foundation of Basic Research, by grant
97-0-13.0-71 from the Russian Ministry of Education, and by the
Austrian Academy of Sciences, "Verwaltungsstelle für
Auslandsbeziehungen."